Gravesande, Willem Jacob 's, Physices elementa mathematica, experimentis confirmata sive introductio ad philosophiam Newtonianam; Tom. 1

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        <div xml:id="echoid-div257" type="section" level="1" n="90">
          <p>
            <s xml:id="echoid-s1630" xml:space="preserve">
              <pb o="34" file="0070" n="73" rhead="PHYSICES ELEMENTA"/>
            que, ut ex ante demonſtratis deducitur; </s>
            <s xml:id="echoid-s1631" xml:space="preserve">ideò ſi ſuſtineatur punctum C,
              <note symbol="*" position="left" xlink:label="note-0070-01" xlink:href="note-0070-01a" xml:space="preserve">129.</note>
            ſtinentur puncta A & </s>
            <s xml:id="echoid-s1632" xml:space="preserve">B, & </s>
            <s xml:id="echoid-s1633" xml:space="preserve">harum actio in puncto C quaſi coacta eſt.</s>
            <s xml:id="echoid-s1634" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1635" xml:space="preserve">Detur tertium punctum grave D, ponderis cujuſcunque; </s>
            <s xml:id="echoid-s1636" xml:space="preserve">jungantur D & </s>
            <s xml:id="echoid-s1637" xml:space="preserve">
              <lb/>
            C, etiam rectâ inflexili, ponderis experti; </s>
            <s xml:id="echoid-s1638" xml:space="preserve">ſitque in hac punctum E, ita de-
              <lb/>
            terminatum, ut EC ſe habeat ad ED, ut pondus puncti D ad ſummam pon-
              <lb/>
            derum punctorum A & </s>
            <s xml:id="echoid-s1639" xml:space="preserve">B.</s>
            <s xml:id="echoid-s1640" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1641" xml:space="preserve">Si A & </s>
            <s xml:id="echoid-s1642" xml:space="preserve">B juncta darentur in C, circa E daretur æquilibrium, poſitâ li-
              <lb/>
            neâ CD in ſitu quocunque : </s>
            <s xml:id="echoid-s1643" xml:space="preserve">ſed A & </s>
            <s xml:id="echoid-s1644" xml:space="preserve">B, ut demonſtravimus, in ſitu
              <note symbol="*" position="left" xlink:label="note-0070-02" xlink:href="note-0070-02a" xml:space="preserve">129</note>
            cunque lineæ AB, agunt quaſi in C juncta eſſent; </s>
            <s xml:id="echoid-s1645" xml:space="preserve">ergo tria pondera A,B,D,
              <lb/>
            lineis inflexilibus conjuncta, in ſitu quocunque, in æquilibrio ſunt circa
              <lb/>
            punctum E; </s>
            <s xml:id="echoid-s1646" xml:space="preserve">quod ergo eſt centrum gravitatis trium punctorum. </s>
            <s xml:id="echoid-s1647" xml:space="preserve">Puncta
              <lb/>
            hæc etiam nullum aliud habere centrum gravitatis, præter punctum E, ex eâ-
              <lb/>
            dem demonſtratione conſtat.</s>
            <s xml:id="echoid-s1648" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1649" xml:space="preserve">Si quartum daretur punctum grave, lineâ inflexili, rectâ, jungendum hoc
              <lb/>
            foret cum E, & </s>
            <s xml:id="echoid-s1650" xml:space="preserve">ſimili demonſtratione conſtaret, quatuor puncta commune
              <lb/>
            habere gravitatis centrum, & </s>
            <s xml:id="echoid-s1651" xml:space="preserve">unicum hoc eſſe.</s>
            <s xml:id="echoid-s1652" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1653" xml:space="preserve">Cum vero eadem demonſtratio ad numerum quemcunque punctorum re-
              <lb/>
            ferri poſſit, applicari poterit omnibus punctis gravibus, ex quibus corpus quod-
              <lb/>
              <note position="left" xlink:label="note-0070-03" xlink:href="note-0070-03a" xml:space="preserve">157.</note>
            cunque conſtat: </s>
            <s xml:id="echoid-s1654" xml:space="preserve">habet ideò corpus centrum gravitatis, & </s>
            <s xml:id="echoid-s1655" xml:space="preserve">unicum tale habet cen-
              <lb/>
            trum.</s>
            <s xml:id="echoid-s1656" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div261" type="section" level="1" n="91">
          <head xml:id="echoid-head138" style="it" xml:space="preserve">De Centri gravitatis inveſtigatione.</head>
          <p>
            <s xml:id="echoid-s1657" xml:space="preserve">Dentur corpora, numero quocunque, quorum commune gravitatis cen-
              <lb/>
              <note position="left" xlink:label="note-0070-04" xlink:href="note-0070-04a" xml:space="preserve">158.</note>
            trum ſit C; </s>
            <s xml:id="echoid-s1658" xml:space="preserve">per hoc concipiamus planum horizontale, quod ſit planum ipſi-
              <lb/>
              <note position="left" xlink:label="note-0070-05" xlink:href="note-0070-05a" xml:space="preserve">TAB. VIII.
                <lb/>
              fig. 2.</note>
            us figuræ. </s>
            <s xml:id="echoid-s1659" xml:space="preserve">Sint centra gravitatis ipſorum corporum A, B, D, E, F; </s>
            <s xml:id="echoid-s1660" xml:space="preserve">ſi
              <lb/>
            centra hæc ipſo plano horizontali memorato non dentur, ad hoc referenda
              <lb/>
            ſunt lineis verticalibus & </s>
            <s xml:id="echoid-s1661" xml:space="preserve">eodem modo planum corpora gravabunt ac ſi i-
              <lb/>
            pſorum centra gravitatis darentur in punctis, in quibus lineæ hæ verticales
              <lb/>
              <note symbol="*" position="left" xlink:label="note-0070-06" xlink:href="note-0070-06a" xml:space="preserve">128.</note>
            planum ſecant .</s>
            <s xml:id="echoid-s1662" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1663" xml:space="preserve">Suſtineatur planum linea GH; </s>
            <s xml:id="echoid-s1664" xml:space="preserve">habentur actiones ponderum ad movendum
              <lb/>
            planum circa lineam GH, multiplicando pondus unumquodque per ſuam diſtanti-
              <lb/>
              <note position="left" xlink:label="note-0070-07" xlink:href="note-0070-07a" xml:space="preserve">159.</note>
            am a linea GH , & </s>
            <s xml:id="echoid-s1665" xml:space="preserve">ſumma productorum dat integram actionem, qua
              <note symbol="*" position="left" xlink:label="note-0070-08" xlink:href="note-0070-08a" xml:space="preserve">131.</note>
            pondera ſimul planum premunt ad hoc circa GH movendum.</s>
            <s xml:id="echoid-s1666" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1667" xml:space="preserve">Omnia autem pondera agunt, quaſi eſſent in C ; </s>
            <s xml:id="echoid-s1668" xml:space="preserve">idcirco habetur etiam
              <note symbol="*" position="left" xlink:label="note-0070-09" xlink:href="note-0070-09a" xml:space="preserve">153.</note>
            pſorum actio, multiplicando ſummam ponderum per diſtantiam puncti C a
              <lb/>
            linea GH: </s>
            <s xml:id="echoid-s1669" xml:space="preserve">Si ergo ſumma memorata productorum, quæ, ut patet, huic ul-
              <lb/>
            timo producto æqualis eſt, dividatur per ſummam ponderum, datur in quotiente
              <lb/>
            diſtantia centri gravitatis a linea GH.</s>
            <s xml:id="echoid-s1670" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1671" xml:space="preserve">Quando agitur de ponderibus, quæ lineis verticalibus ad planum horizon-
              <lb/>
            tale referuntur, diſtantiæ punctorum, ad quæ pondera referuntur, à lineâ
              <lb/>
            GH, ſunt æquales diſtantiis centrorum gravitatis ipſorum corporum à pla-
              <lb/>
            no verticali, per GH tranſeunti.</s>
            <s xml:id="echoid-s1672" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1673" xml:space="preserve">Cum verò hæc demonſtratio locum habeat in quocunque ſitu corpora den-
              <lb/>
            tur, ſi lineis inflexilibus, & </s>
            <s xml:id="echoid-s1674" xml:space="preserve">ſine pondere, corpora inter ſe coh@reant, nul-
              <lb/>
            lum poteſt concipi planum, quod non, ſervato ipſius ſitu reſpectu corporum,
              <lb/>
              <note position="left" xlink:label="note-0070-10" xlink:href="note-0070-10a" xml:space="preserve">160.</note>
            poſſit fieri verticale; </s>
            <s xml:id="echoid-s1675" xml:space="preserve">unde ſequitur datis corporibus & </s>
            <s xml:id="echoid-s1676" xml:space="preserve">plano quocunque, diſtan-
              <lb/>
            tiam centri gravitatis a plano detegi, multiplicando corpus unumquodque per ſui
              <lb/>
            centri gravitatis diſtantiam a plano, & </s>
            <s xml:id="echoid-s1677" xml:space="preserve">dividendo productorum ſummam per ipſo-
              <lb/>
            rum corporum ſummam.</s>
            <s xml:id="echoid-s1678" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1679" xml:space="preserve">Si ſimilem demonſtrationem applicemus plano, quod inter corpora tranſit,
              <lb/>
              <note position="left" xlink:label="note-0070-11" xlink:href="note-0070-11a" xml:space="preserve">161.</note>
            differentia inter ſummas productorum ab utraque parte per corporum ſum-</s>
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